Subgroups — #13092453

Order 1472.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#117	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10624990	C₂	1	2	yes	maximal, complemented
5	#10624989	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂²⋊C₄×C₂₃	—	2	1	no	—
9	C₄²×C₂₃	—	2	1	no	—
10	C₂²⋊Dic₂₃	—	2	1	no	—
11	C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₄×Dic₂₃	—	2	1	no	—
13	C₂×C₂₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂³×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	—	2	2	no	—
18	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	Q₈	1	2	yes	—
21	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₂₃	—	2	2	no	—
23	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
24	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₂₃	—	2	2	no	—
26	C₂²×C₂₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₃	—	4	2	no	—
29	C₄×C₂₃	C₄⋊C₄	1	2	yes	—
30	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
32	Dic₂₃	—	4	2	no	—
33	#427	—	23	1	no	maximal, Sylow
34	C₂×C₂₃	—	2	2	no	—
35	C₂×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₂₃	#135	1	2	yes	—
37	C₂²×C₈	—	23	1	no	—
38	#117	—	23	1	no	—
39	#117	D₂₃	1	1	yes	characteristic
40	#107	—	23	2	no	—
41	#106	—	23	2	no	—
42	C₂₃	#427	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
44	C₂×C₈	—	23	4	no	—
45	C₂×C₈	—	23	1	no	—
46	C₂×C₈	—	23	1	no	—
47	C₄⋊C₄	—	23	2	no	—
48	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	46	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	46	1	no	—
53	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	46	2	no	—
56	C₂×C₄	Dic₄₆	1	2	yes	—
57	C₂×C₄	C₄×D₂₃	1	2	yes	—
58	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
60	C₈	—	23	2	no	—
61	C₈	—	23	2	no	—
62	C₂²	—	2	2	no	—
63	C₂²	Dic₂₃⋊C₄	1	2	yes	—
64	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
65	C₄	—	4	2	no	—
66	C₄	—	92	2	no	—
67	C₄	Dic₂₃⋊C₄	1	2	yes	—
68	C₄	C₂×Dic₄₆	1	1	yes	characteristic
69	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
72	C₂	#10625121	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#117	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂³×C₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#117	D₂₃
hypocenter	C₂₃	#427
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #10624990	C₂	semidirect
5. #10624989	C₂	semidirect
13. C₂×C₂₃⋊C₈	C₄	semidirect
42. C₂₃	#427	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#117	2
#10625097	2
C₂²×C₂₃⋊C₈	2
#10624990	2
#10624989	2
#427	23